Application of projection algorithms to differential equations: boundary value problems

TL;DR

This paper reformulates second order boundary value problems as feasibility problems involving hypersurfaces and applies the Douglas-Rachford method, demonstrating stability and parallelization advantages over traditional methods.

Contribution

It introduces a novel reformulation of boundary value problems as feasibility problems on up to three sets, enabling stable and parallelizable solutions.

Findings

Douglas-Rachford method is stable for BVPs where Newton's method fails

Reformulation allows parallel computation of solutions

Method effective on various boundary value problem examples

Abstract

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails.